As for conversion in general, the S.O. and I have lived all over the country, including multiple major cities (Tucson, Atlanta, Washington, D.C.). We've had a lot of different "missionaries" try to convert us. Mormons, Jehovah's Witnesses, Scientologists, Pentecostals, Baptists, even a few Buddhists. The weirdest one was probably The Process Church of the Final Judgement -- they always reminded me of the Satanist branch of the Salvation Army. Maybe it was the uniforms. Anyway, the one religion that has never in my lifetime tried to persuade me that I should take a look at it is Islam. I've known quite a few Muslims -- co-workers, college friends, students, neighbors, a landlord -- and not once has any of them ever shoved a Koran in my face and told me I need to read the good news about God. Both of my kids had good friends who were Muslim; those girls were the only ones who never tried to persuade my kids to attend religious services with them. We had Baptists, Lutherans, and Pentecostals all trying to lure my kids away from whatever church we might attend to go to their church services instead, but the Muslims apparently figured your religion was your own business.
I know the Nation of Islam, which is a thoroughly American institution, does missionary work in prisons and in the African-American community. We've lived in cities where Nation of Islam members can be seen selling their newspaper on the street. But you don't see Nation of Islam members going door to door ringing doorbells and trying to persuade you to let them into your house to discuss the Koran with you, unlike the Jehovah's Witnesses and the Mormons who are so aggressive in their
Yes, it's true that historically Islam did conversion by the sword. But so did Christianity. If you look back a few centuries, both religions did the "convert or die." I have, in fact, heard a few nominal Christians suggest that's what we should do now -- a few years ago, right-wing spokesperson Ann Coulter said basically that: tell Muslims to convert to Christianity and if they don't, kill them -- but that's not exactly mainstream thinking.
In short, I'm not sure where the conversion fear comes from because it's surely not from being harassed by street corner imams passing out pamphlets and telling us to give up our sinful Western ways. It also can't be from observing Da'esh indulging in LARPing in the Syrian desert because thus far Da'esh seems more interested in terrorizing and chasing out non-Muslims than they do in converting anyone. Several million refugees fleeing the region doesn't exactly suggest that Da'esh is engaging in any meaningful proselyzation efforts. It's a mystery.